Citation: Kosakovsky Pond S, Delport W, Muse SV, Scheffler K (2010) Correcting the Bias of Empirical Frequency Parameter Estimators in Codon Models. PLoS ONE 5(7):
e11230.
https://doi.org/10.1371/journal.pone.0011230
Editor: Thomas Mailund, Aarhus University, Denmark
Received: April 6, 2010; Accepted: June 1, 2010; Published: July 30, 2010
Copyright: © 2010 Kosakovsky Pond et al. This is an open-access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.
Funding: This research was supported by the Joint Division of Mathematical Sciences/National Institute of General Medical Sciences Mathematical Biology Initiative through Grant NSF-0714991, the National Institutes of Health, AI47745 and by a University of California, San Diego Center for AIDS Research/NIAID Developmental Award to S.L.K.P. (AI36214). The funders had no role in study design, data collection and analysis, decision to publish, or preparation of the manuscript.
Competing interests: The authors have declared that no competing interests exist.
Introduction
Virtually all codon models in wide use today (see [1], [2] for recent reviews) are members of the class of finite-state, continuous time reversible Markov chains, each defined by an instantaneous rate matrix 
. Transition matrices for finite amounts of time are found via the matrix exponential of 
, so the probability that a position initially occupied by codon 
is occupied by codon 
after 
units of time is 
(throughout the manuscript we will use upper-case letters to index codons and lower-case letters to index nucleotides). If 
is a model in this class, the individual entries of its rate matrix can be written in the canonical form 
. The 
can be thought of as “rate parameters” that govern the relative rates of substitutions between different codons, while parameters 
induce the equilibrium frequencies of the codons. The choice of 
is the primary distinction between the two popular families of codon models: MG (introduced in [3]) and GY (introduced in [4]). How to best estimate the 
— or more precisely, how to estimate model parameters that actually determine the 
— from sequence alignments is the focus of this note. In order to frame this discussion we need to define what we mean by empirical frequencies, model parameters and equilibrium frequencies (Figure 1). Given an observed alignment, the position-specific empirical nucleotide frequencies, 
where 
is a nucleotide (
) and 
the codon position (
), can be estimated directly by counts from the data, and the empirical codon frequencies, 
, can be estimated by counts as well (the latter gives rise to the F61 codon frequency estimator [4]). Either of these estimates can be used to set model parameters, however typical alignments have insufficient information for the direct estimation of empirical codon frequencies with a sufficient degree of confidence. Rather, the empirical nucleotide frequencies are used to set the nucleotide frequency parameters, 
, and by multiplication of their constituents, the codon frequency parameters, 
. For example, in the original MG94 model of codon evolution [3], the equilibrium frequency of codon 
is given by 
, where 
. A common extension of this model, referred to as MG94 F3×4, allows the three codon positions to have their own nucleotide frequency parameters and leads to equilibrium codon expressed as:
(1)In this expression the superscripts indicate the position, and the equation for 
is modified in the obvious way. If we set all of the model nucleotide frequency parameters to be equal, i.e. 
, the result is equal equilibrium frequencies for all codons, i.e. 
for all 
. This vector of codon equilibrium frequencies allows us to easily tabulate, via marginalization, the equilibrium frequencies of each nucleotide at each position:
(2)
Note that there are only 13 occurrences of T in the first position, 14 of A in the second position, etc because the model explicitly disallows (TAG,TAA,TGA) as is standard for all other codon models. The finding from this exercise is that when one sets all the 
, each of the codon equilibrium frequencies, 
takes the anticipated value of 
. However, remarkably, the equilibrium nucleotide frequencies generated by this model are not the anticipated 
. For instance, the equilibrium frequency of 
at the first position is 
. Traditionally, the empirical nucleotide frequencies are used to set nucleotide frequency parameters, and it is therefore assumed that the induced equilibrium nucleotide frequencies are equal to those observed in the alignment. However, given that the nucleotide composition of stop codons is not accounted for, this practice is flawed, because 
. The conflation of frequency parameters (
) and equilibrium nucleotide (
) frequencies results in incorrect estimates of equilibrium nucleotide (and codon) frequencies as demonstrated in (2) above. This phenomenon is not restricted to the MG family of models. It is simple to demonstrate the exact same behavior for the GY family of models, again because of the incorrect designation of nucleotide frequency parameters in the rate matrix as equal to empirical nucleotide frequencies. We show that the traditional identification of frequency parameters and observed nucleotide frequencies leads to a cascade of problems. Model frequency parameters are estimated with bias, which leads to biased estimation of the equilibrium codon frequencies, which leads to compensatory biased estimation of the substitution rate parameters. We propose a correction, and a maximum likelihood frequency parameterization and show that both these approaches are not similarly biased, and therefore advocate their use in codon models.
Materials and Methods
To ensure clarity of presentation, we first carefully introduce the necessary notation (summarized in Figure 1). For a given substitution model, let 
be the frequency of sense codon 
(
) in its equilibrium distribution, and 
, 
be the equilibrium frequency of nucleotide 
in codon position 
. When necessary, we will indicate specific models via a superscript (ie, MG or GY). The position specific nucleotide equilibrium frequencies, 
, are uniquely determined by the codon equilibrium frequencies, 
, through marginalization, e.g. 
is simply the sum of frequencies of the 
sense codons that have a T in their first position, e.g. as in equation (2).
These equilibrium frequencies, of both nucleotides and codons, have traditionally been assumed equal to empirical frequencies observed in a sequence alignment, 
or 
, and used to set model parameters. If the specified model is correct, 
converges to 
and 
to 
as the sequence length 
increases. (However, note that this result requires that the evolutionary process itself be at equilibrium; many important biological mechanisms— notably directional positive selection— are likely to disrupt equilibrium; see [5]–[7]).
Because the simple example in equation (2) demonstrated that the empirical and equilibrium nucleotide frequencies are not synonymous, we strive to obtain an expression that relates the equilibrium nucleotide frequencies to the model nucleotide frequencies, 
, and through extension –to the observed empirical frequencies. Even though the MG and GY models treat equilibrium codon frequencies differently, it is a fortunate coincidence that in either case the 
have identical forms when written in terms of 
. Given twelve MG nucleotide frequency parameters, only 
of which are independent because 
for each position 
, the equilibrium frequency of codon 
induced by their values is as in equation (1).
By using 
to directly estimate 
in equation (1), one obtains the popular 
estimator of codon equilibrium frequencies – by far the most common estimator used in literature for both MG and GY classes of models. The statistical and computational appeal of 
lies in its use of only 
nucleotide parameters to describe 
codon frequencies. However, the key shortcut— direct estimation of nucleotide frequency parameters with empirical nucleotide frequencies from the data— is flawed. The empirical nucleotide frequencies are unbiased estimates of the true equilibrium frequencies; unfortunately, the model parameters they are being used to estimate are something different. Thus, a fundamental problem with current practices is that use of the 
estimators with either MG or GY models leads to biased estimates of the 
, and in turn the 
. As we will show below, the problems do not end there, and lead to biased estimation of other model parameters.
We first present two approaches for correcting these estimation errors. The obvious, but more computationally demanding method is to estimate the 
by maximum likelihood along with other model parameters. We dub this approach 
. Theory suggests that estimates from this methodology will have all the desirable properties of maximum likelihood estimation. Maximum likelihood estimation of these values has been available in some software packages, e.g. in HyPhy [8], for a number of years, but to our knowledge it has rarely been used.
The second strategy, described here for the first time, relies on finding an expression for the induced equilibrium frequency of nucleotide 
at codon position 
(
) as a function of 
. Since the 
define codon equilibrium frequencies (equation 1), we can readily obtain such equations by marginalization:
(3)
Here, 
is simply scaling for the absence of stop codons: 
, and 
defines the set of stop codons. The corrected 
, or 
estimator equates 
with observed nucleotide frequencies 
, and then solves the nonlinear system (3) for 
to obtain estimates of the latter. Because 
, the above system of 
nonlinear equations relate 
independent observed statistics (
, e.g. for 
) with 
independent model parameters 
. We were unable to obtain a closed form solution to the system, but it can be easily solved numerically at a negligible computational cost.
We conducted simulations to further investigate the effects of biases in the equilibrium frequencies on parameters typically estimated using phylogenetic models. We generated two-sequence codon alignments with uniform codon frequency composition (
). We used 
as substitution bias parameters in the MG94xREV model [9], and set the nonsynonymous/synonymous substitution rate ratio 
to 
. The two sequences were 
divergent on average, and the length of the alignment, 
, was one of 
, 
or 
codons. 
replicates were generated for each value of 
. We compared the fits of 
, 
and 
on simulated data sets, and furthermore compared simulated to inferred parameter estimates with each of the three frequency parameterizations. In addition to the simulated data, we fitted all three frequency parameterizations to a sample of 
alignments from the carefully curated Pandit database [10]. All alignments were chosen to contain between 10 and 20 sequences and at least 200 reliably aligned codon sites. Given that each estimator has the same number of independent parameters (
), an improvement in log-likelihood under one of the models is considered as evidence in favor of the better fitting model, e.g. under the BIC [11] criterion. All new estimators for the MG94 class of models are implemented in HyPhy.
Results and Discussion
We simulated data with a uniform codon frequency composition and fitted all three frequency parameterizations for alignments of various sequence lengths. The suboptimal nature of the 
estimator is immediately apparent from Figure 2a, where the improvement in 
scores of the model equipped with the corrected estimator 
is shown. For all replicates, the 
estimator yielded better 
, with median improvements of 
, 
, and 
(for 
, and 
codons respectively), or approximately 
likelihood points per codon site. Note that as the sample size increased, the estimators from (3) effectively matched the performance of the maximum likelihood estimator (Figure 2b). Even more importantly, the use of the 
frequency estimator led to biased inference of other model parameters. Maximum likelihood estimates of some substitution rates were biased under the 
, and the bias was progressively more pronounced with increasing sample size (Figure 2c). Indeed, for 
, a simple likelihood ratio test rejected the (true) null of 
at 
for all 
replicates. Biased MLEs of the substitution rate parameter 
is a result of the under/overestimates of 
and 
using 
. Similar results were seen for the other 
. To our relief, the maximum likelihood estimate (MLE) for the 
ratio was not noticeably affected even for the largest sample size (mean 
, median 
, IQR 
under 
; mean 
, median 
, IQR 
under 
, Figure 2d).
For the Pandit alignments 
values were, of course, higher for the models estimated using 
than for those using 
. However, the magnitudes of the differences were impressive (median 
, IQR 
, max 
). The 
estimator improved the 
score of the 
estimator for over 
(
) of the alignments by a median of 
points; in the remaining cases the median decrease in 
score was 
points. As with the simulated data, the MLEs of 
were largely unaffected by the choice of frequency estimators (but there were some datasets where the difference was large), while some substitution rate estimates appeared biased (Figure 3). For example, the estimates of 
were strongly linearly correlated between 
and 
methods (
), but the regression line was estimated as 
, which recapitulates the downward bias observed on simulated data (if the estimates were unbiased, we would expect an intercept of zero and slope of one).
We have demonstrated through simulations that the almost universally used 
estimator of equilibrium frequencies in codon substitution models is biased, and we have pointed out how a misinterpretation of standard codon model parameters is responsible for these biases. Although this bias appears to have little effect on estimation of “composite” parameters such as the nonsynonymous/synonymous rate ratio (
) and branch lengths (results not shown), the bias has considerable damaging effects on the estimation of substitution rate parameters in the instantaneous rate matrix. This problem will become acutely relevant as researchers pursue finer-scale studies of the evolutionary process, such as developing substitution models with protein residue-dependent codon substitution rates [12], [13]. Since the computational burden of the 
estimator is virtually identical to that of our proposed 
estimator, which in turn is only marginally faster than 
, we recommend the use of either of the alternatives offered in this manuscript over the 
estimator. Our current recommendation is to obtain 
estimates and use them to initialize the optimization procedure for 
to speed up convergence.
Author Contributions
Conceived and designed the experiments: SLKP WD SVM KS. Performed the experiments: SLKP. Analyzed the data: SLKP. Contributed reagents/materials/analysis tools: SLKP. Wrote the paper: SLKP WD SVM KS.
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